On the geometry of quantum indistinguishability

An algebraic approach to the study of quantum mechanics on configuration spaces with a finite fundamental group is presented. It uses, in an essential way, the Gelfand-Naimark and Serre-Swan equivalences and thus allows one to represent geometric properties of such systems in algebraic terms. As an application, the problem of quantum indistinguishability is reformulated in the light of the proposed approach. Previous attempts aiming at a proof of the spin-statistics theorem in non-relativistic quantum mechanics are explicitly recast in the global language inherent to the presented techniques. This leads to a critical discussion of single-valuedness of wave functions for systems of indistinguishable particles. Potential applications of the methods presented in this paper to problems related to quantization, geometric phases and phase transitions in spin systems are proposed.Comment: 24 page

Spatial Besov Regularity for Stochastic Partial Differential Equations on Lipschitz Domains

We use the scale of Besov spaces B^\alpha_{\tau,\tau}(O), \alpha>0, 1/\tau=\alpha/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains O\subset R^d. The Besov smoothness determines the order of convergence that can be achieved by nonlinear approximation schemes. The proofs are based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions.Comment: 32 pages, 3 figure

Remarks on the Configuration Space Approach to Spin-Statistics

The angular momentum operators for a system of two spin-zero indistinguishable particles are constructed, using Isham's Canonical Group Quantization method. This mathematically rigorous method provides a hint at the correct definition of (total) angular momentum operators, for arbitrary spin, in a system of indistinguishable particles. The connection with other configuration space approaches to spin-statistics is discussed, as well as the relevance of the obtained results in view of a possible alternative proof of the spin-statistics theorem.Comment: 18 page

Similarity and bisimilarity notions appropriate for characterizing indistinguishability in fragments of the calculus of relations

Motivated by applications in databases, this paper considers various fragments of the calculus of binary relations. The fragments are obtained by leaving out, or keeping in, some of the standard operators, along with some derived operators such as set difference, projection, coprojection, and residuation. For each considered fragment, a characterization is obtained for when two given binary relational structures are indistinguishable by expressions in that fragment. The characterizations are based on appropriately adapted notions of simulation and bisimulation.Comment: 36 pages, Journal of Logic and Computation 201

Some views on information fusion and logic based approaches in decision making under uncertainty

Decision making under uncertainty is a key issue in information fusion and logic based reasoning approaches. The aim of this paper is to show noteworthy theoretical and applicational issues in the area of decision making under uncertainty that have been already done and raise new open research related to these topics pointing out promising and challenging research gaps that should be addressed in the coming future in order to improve the resolution of decision making problems under uncertainty